CFD: Cost-aware Fusion-based Decomposition

• Cost-aware Fusion-based Decomposition (CFD) is a heuristic framework that leverages local Clifford equivalence to transform target graph states into synthesis-friendly representations.
• It decomposes graph states into ring, star, and linear motifs, strategically reducing the number of probabilistic Type-I fusion operations and resource overhead.
• Empirical evaluations show up to 84.6% reduction in photon overhead and multi-order improvements in generation rates relative to baseline photonic construction methods.

Searching arXiv for the target paper and closely related graph-state synthesis work to ground the article. Cost-aware Fusion-based Decomposition (CFD) is a heuristic framework for photonic graph-state synthesis that treats preparation of a target graph state as a cost-sensitive decomposition-and-assembly problem under probabilistic fusion. Introduced for photonic graph states synthesized with Type-I fusion operations, CFD first exploits local Clifford (LC) equivalence to replace a target graph $G$ by a more synthesis-friendly LC-equivalent representative $G'$, then decomposes $G'$ into directly preparable ring, star, and linear motifs, and finally assembles those motifs through Type-I fusion to reduce fusion overhead and physical-qubit consumption (Ji et al., 1 Jun 2026). The framework is explicitly motivated by the exponential degradation of photonic generation rate with increasing fusion count, and its empirical evaluation reports up to $84.6\%$ reduction in resource overhead relative to baseline constructions, together with generation-rate improvements spanning multiple orders of magnitude (Ji et al., 1 Jun 2026).

1. Conceptual setting and synthesis objective

CFD is defined in the setting of photonic graph states represented by a simple undirected graph $G=(V,E)$, where vertices denote photonic qubits and edges denote entangling $CZ$ relations (Ji et al., 1 Jun 2026). The associated graph state is written as

$$|G\rangle = \prod_{(u,v)\in E} CZ_{u,v}\;|+\rangle^{\otimes |V|}.$$

In an abstract gate model, preparation would proceed by applying $CZ$ gates on all edges, but the framework is designed for photonic platforms in which scalable deterministic two-qubit entangling gates are not assumed and synthesis instead relies on directly generable multipartite resource states plus probabilistic fusion (Ji et al., 1 Jun 2026).

The cost sensitivity arises from the use of Type-I fusion with success probability $1/2$. Each additional successful fusion contributes both a $(1/2)^k$ success-probability penalty and an additional photon-loss factor $G'$0 in the adopted rate model, so synthesis strategies with many fusion steps suffer an exponential reduction in effective generation rate (Ji et al., 1 Jun 2026). Accordingly, the paper frames the synthesis problem around three linked goals: reducing fusion overhead, reducing physical-qubit or photon consumption, and improving photonic generation rate (Ji et al., 1 Jun 2026).

This suggests that CFD is not merely a graph partitioning procedure. Its purpose is to choose a structurally favorable representation of the target entanglement resource and then realize that representation using a motif vocabulary aligned with hardware-native photonic resource states (Ji et al., 1 Jun 2026).

2. LC equivalence as a preprocessing mechanism

A central feature of CFD is the use of LC equivalence before decomposition. Two graph states are LC-equivalent if one graph can be transformed into the other by a sequence of local complementations, where local complementation at a vertex $G'$1 complements the induced subgraph on the neighborhood $G'$2 (Ji et al., 1 Jun 2026). Formally,

$G'$3

with $G'$4 denoting symmetric difference (Ji et al., 1 Jun 2026). Because LC operations correspond to local single-qubit Clifford gates, they preserve the entanglement resource up to local operations, so an implementation may synthesize any LC-equivalent representative and then recover the target by inexpensive local corrections (Ji et al., 1 Jun 2026).

Within CFD, LC preprocessing is used to select an overview-friendly representative, preferably the member of the LC class with the minimum number of edges (Ji et al., 1 Jun 2026). For $G'$5 to $G'$6, the framework uses a precompiled graph-state orbit dataset to identify the minimum-edge representative directly; for $G'$7, it points to an external method from prior work for finding an LC-equivalent graph with fewer edges (Ji et al., 1 Jun 2026).

The paper treats minimum-edge selection as a practical proxy rather than a theorem of optimality. Empirically, however, it reports that CFD applied to the minimum-edge representative exactly matches the best CFD result over the full LC class for all tested classes up to $G'$8, and remains close for most larger classes (Ji et al., 1 Jun 2026). The reported generation-rate gains of CFD(min-edge) over CFD(target) increase sharply with graph size: about $G'$9–$G'$0 at $G'$1, about $G'$2–$G'$3 at $G'$4, up to $G'$5 at $G'$6, over $G'$7 at $G'$8, over $G'$9 at $84.6\%$0, and nearly $84.6\%$1 at $84.6\%$2 (Ji et al., 1 Jun 2026).

A plausible implication is that LC preprocessing is the dominant structural lever in regimes where the fusion penalty compounds rapidly, especially as graph size increases.

3. Motif library and three-stage decomposition procedure

CFD uses a restricted motif library consisting of star, linear, and ring motifs (Ji et al., 1 Jun 2026). A star motif is a GHZ-type graph on $84.6\%$3 qubits with one central vertex connected to $84.6\%$4 leaves; a linear motif is a path graph on $84.6\%$5 vertices; and a ring motif is a cycle graph on $84.6\%$6 vertices (Ji et al., 1 Jun 2026). The decomposition is an edge-disjoint cover,

$84.6\%$7

so vertices may be shared but each edge is assigned to exactly one motif (Ji et al., 1 Jun 2026).

The framework is explicitly organized as a three-stage heuristic pipeline (Ji et al., 1 Jun 2026):

Stage	Operation	Output
1	LC preprocessing	Synthesis-friendly $84.6\%$8
2	Motif decomposition	Edge-disjoint ring, star, linear motifs
3	Fusion-based assembly	Photonic realization of $84.6\%$9

In the decomposition stage, motif extraction follows the order ring $G=(V,E)$0 star $G=(V,E)$1 linear (Ji et al., 1 Jun 2026). Ring extraction repeatedly searches for ring templates using VF2 subgraph isomorphism, strips matched ring edges from the residual graph, and removes isolated vertices (Ji et al., 1 Jun 2026). The worst-case complexity for a ring motif $G=(V,E)$2 of size $G=(V,E)$3 is given as $G=(V,E)$4 per match attempt, and repeated extraction over templates yields

$G=(V,E)$5

where $G=(V,E)$6 is the number of matching attempts (Ji et al., 1 Jun 2026).

After ring removal, star motifs are extracted by a degree-based heuristic. For each edge $G=(V,E)$7, the higher-degree endpoint is assigned as the star center; ties are broken by smaller vertex index; incident edges are aggregated into candidate star sets $G=(V,E)$8; and a star is retained only if $G=(V,E)$9 (Ji et al., 1 Jun 2026). This stage has linear complexity,

$CZ$0

and is intended to prevent hub structures from being fragmented into multiple path motifs (Ji et al., 1 Jun 2026).

The residual graph is then decomposed into linear motifs, with residual cycles reclassified as rings (Ji et al., 1 Jun 2026). For a connected component $CZ$1, if all degrees are 2, it is decomposed into simple cycles. Otherwise, odd-degree vertices are paired arbitrarily, virtual edges are added to produce an Eulerian multigraph, an Eulerian circuit is computed, and the resulting traversal is cut at virtual edges and repeated vertices until all motifs are vertex-simple (Ji et al., 1 Jun 2026). This Eulerization-based stage also runs in

$CZ$2

assuming adjacency-list implementation and use of Hierholzer’s algorithm (Ji et al., 1 Jun 2026).

The paper provides an explicit rationale for the extraction order. Rings are removed first because cycles are self-contained and should not be fragmented into paths; stars precede linear extraction because otherwise a hub of degree $CZ$3 may be split into $CZ$4 separate path motifs instead of one star (Ji et al., 1 Jun 2026). Over minimum-edge representatives for graph states up to $CZ$5, the authors report that ring $CZ$6 star $CZ$7 linear has the lowest mean resource overhead at every tested graph size (Ji et al., 1 Jun 2026).

4. Cost model and optimization criteria

The cost-aware aspect of CFD is formalized through the photonic generation-rate model. For an $CZ$8-photon graph state prepared without fusion, the paper uses

$CZ$9

where $$|G\rangle = \prod_{(u,v)\in E} CZ_{u,v}\;|+\rangle^{\otimes |V|}.$$0 is the single-photon generation rate and $$|G\rangle = \prod_{(u,v)\in E} CZ_{u,v}\;|+\rangle^{\otimes |V|}.$$1 is the overall photon collection efficiency (Ji et al., 1 Jun 2026). If synthesis requires $$|G\rangle = \prod_{(u,v)\in E} CZ_{u,v}\;|+\rangle^{\otimes |V|}.$$2 successful Type-I fusions, the effective rate becomes

$$|G\rangle = \prod_{(u,v)\in E} CZ_{u,v}\;|+\rangle^{\otimes |V|}.$$3

Because $$|G\rangle = \prod_{(u,v)\in E} CZ_{u,v}\;|+\rangle^{\otimes |V|}.$$4 is fixed for a given target and the multiplicative penalties decrease monotonically with $$|G\rangle = \prod_{(u,v)\in E} CZ_{u,v}\;|+\rangle^{\otimes |V|}.$$5, the optimization is reduced to minimizing the number of required fusions (Ji et al., 1 Jun 2026): $$|G\rangle = \prod_{(u,v)\in E} CZ_{u,v}\;|+\rangle^{\otimes |V|}.$$6

The paper also evaluates a direct photon-overhead metric

$$|G\rangle = \prod_{(u,v)\in E} CZ_{u,v}\;|+\rangle^{\otimes |V|}.$$7

with motif cost defined as $$|G\rangle = \prod_{(u,v)\in E} CZ_{u,v}\;|+\rangle^{\otimes |V|}.$$8 for ring motifs and $$|G\rangle = \prod_{(u,v)\in E} CZ_{u,v}\;|+\rangle^{\otimes |V|}.$$9 for star and linear motifs (Ji et al., 1 Jun 2026). The extra $CZ$0 for a ring reflects the assumption that generating an $CZ$1-ring requires first preparing an $CZ$2-qubit linear graph and then fusing its endpoints (Ji et al., 1 Jun 2026).

The framework therefore uses two related but distinct notions of cost: fusion count and photon overhead. The paper states that these metrics generally correlate but are not identical, because ring motifs can trade reduced fusion count for slightly increased motif-preparation overhead (Ji et al., 1 Jun 2026). It does not provide a closed-form expression for $CZ$3 in terms of motif count, and no such relation is asserted (Ji et al., 1 Jun 2026).

This suggests that CFD’s practical optimization target is an overview surrogate: minimize fusion count while monitoring resource overhead $CZ$4 as a physically meaningful secondary indicator.

5. Fusion-based assembly and graph-level action of Type-I fusion

After decomposition, each motif is assumed to be independently preparable as a photonic resource state and then connected using Type-I fusion operations (Ji et al., 1 Jun 2026). At the graph level, if $CZ$5 and $CZ$6 are the chosen fusion qubits, successful Type-I fusion removes $CZ$7, keeps $CZ$8, and transfers the adjacency of $CZ$9 to $1/2$0, yielding

$1/2$1

where $1/2$2 denotes the edges incident to $1/2$3 and $1/2$4 denotes the neighborhood of $1/2$5 in $1/2$6 (Ji et al., 1 Jun 2026). On failure, $1/2$7 is measured in the $1/2$8-basis and removed together with its incident edges, while $1/2$9 remains intact (Ji et al., 1 Jun 2026). The failure model therefore implies regeneration and retry of the disconnected fragments.

Several assumptions are explicit in this assembly model (Ji et al., 1 Jun 2026). The Type-I fusion success probability is fixed at $(1/2)^k$0; failures incur regeneration overhead; LC corrections after motif assembly are treated as negligible-cost local operations; and the motifs are taken to be independently preparable. The framework does not provide an explicit algorithm for selecting attachment points or dangling qubits for motif interconnection, nor does it develop a routing or scheduling optimization over fusion order (Ji et al., 1 Jun 2026).

A common misconception is to interpret “fusion” in CFD as a latent or end-to-end learned fusion strategy analogous to multimodal neural architectures. In this work, fusion is literal photonic state fusion through a hardware-defined operation with probabilistic success and graph-theoretic update semantics (Ji et al., 1 Jun 2026).

6. Empirical performance, baselines, and structure dependence

The empirical evaluation of CFD uses three settings: the Graph State Orbits Dataset for graph states with $(1/2)^k$1 to $(1/2)^k$2, two-dimensional lattice or cluster states from $(1/2)^k$3 to $(1/2)^k$4, and three-dimensional cubic lattice graph states of varying sizes (Ji et al., 1 Jun 2026). Generation-rate simulations use $(1/2)^k$5 GHz and $(1/2)^k$6, chosen to model deterministic single-photon emission from InAs/GaAs quantum dots coupled to photonic cavities (Ji et al., 1 Jun 2026).

For orbit data, each target is evaluated as CFD(target), CFD(min-edge), and CFD(best), where the last exhaustively applies CFD to every graph in the LC class and reports the best result (Ji et al., 1 Jun 2026). The principal conclusion is that CFD(min-edge) always outperforms CFD(target), and the advantage grows with graph size (Ji et al., 1 Jun 2026).

CFD is compared against two baselines (Ji et al., 1 Jun 2026). The first is edge-decorated complete graph (EDCG), which has EPR cost $(1/2)^k$7 and resource overhead $(1/2)^k$8. The second is direct edge-based construction (DEBC), with resource overhead $(1/2)^k$9. Relative to EDCG, CFD reports overhead reductions of approximately $G'$00, $G'$01, $G'$02, $G'$03, $G'$04, and $G'$05 for $G'$06 (Ji et al., 1 Jun 2026). Relative to DEBC, the reductions are approximately $G'$07, $G'$08, $G'$09, $G'$10, $G'$11, and $G'$12 over the same sizes (Ji et al., 1 Jun 2026). These reductions are associated with generation-rate improvements spanning multiple orders of magnitude (Ji et al., 1 Jun 2026).

The lattice experiments complicate any universal interpretation of the default extraction order. For highly regular 2D cluster states, no single motif ordering dominates all sizes, and for larger clusters the linear-only policy performs best, reducing overhead by up to about $G'$13 relative to other policies (Ji et al., 1 Jun 2026). For 3D cubic lattices, linear-only becomes best as size increases, with about $G'$14–$G'$15 lower overhead than policies that extract stars or rings early (Ji et al., 1 Jun 2026).

This indicates that CFD’s default policy is strong on average over irregular graph-state orbit data but not universally optimal for highly regular lattices (Ji et al., 1 Jun 2026). A plausible implication is that motif policy should itself be made structure-adaptive if the framework is extended to broader graph families.

7. Significance, limitations, and relation to adjacent “cost-aware decomposition” work

CFD is best understood as a hardware-aware graph-state compiler specialized to photonic synthesis under probabilistic Type-I fusion, with a restricted motif library and negligible-cost local Clifford corrections (Ji et al., 1 Jun 2026). Its main conceptual contribution is the joint use of LC equivalence and motif decomposition under a photonic cost model. The framework is therefore neither only an LC-orbit search nor only a motif cover heuristic; it combines both under an overview objective tied to experimentally motivated rate penalties (Ji et al., 1 Jun 2026).

Several limitations are explicit or strongly implied in the evaluation (Ji et al., 1 Jun 2026). The framework is heuristic and not guaranteed optimal. Its motif library is restricted to ring, star, and linear motifs. LC search beyond $G'$16 is delegated to external methods. Attachment-point selection and detailed fusion scheduling are abstracted away. The cost model neglects nonidealities associated with LC corrections and does not fully specify assembly-level fusion count from motifization. Finally, the lattice experiments show that the default extraction order is not universally best (Ji et al., 1 Jun 2026).

The phrase “Cost-aware Fusion-based Decomposition” also appears in other research contexts with different meanings, which can create terminological confusion. In topology-aware data repair, “CFD” denotes conditional functional dependency rather than this photonic synthesis framework (Zhao et al., 27 Jan 2026). In GPU kernel planning, related ideas appear as cost-aware operator fusion guided by memory-access models, but without LC equivalence or photonic synthesis semantics (Qararyah et al., 2024). In LLM-based automation for computational fluid dynamics, decomposition and empirical cost trade-offs are studied, but “CFD” refers to computational fluid dynamics rather than Cost-aware Fusion-based Decomposition (Chen et al., 1 Feb 2025, Wang et al., 2 Apr 2025). These adjacent usages highlight that the photonic-graph-state meaning of CFD is domain-specific and should not be conflated with homonymous acronyms in data cleaning or fluid simulation.

Within photonic graph-state synthesis itself, the distinctive claim of CFD is that the minimum-edge LC representative is a highly effective proxy for near-optimal synthesis when combined with structure-aware motif decomposition (Ji et al., 1 Jun 2026). That claim is empirical rather than formal, but it is the framework’s most consequential message for practice. It suggests that scalable synthesis gains can be obtained without exhaustive LC-orbit search, provided that one exploits LC sparsification before decomposition and matches the decomposition vocabulary to hardware-realizable motifs (Ji et al., 1 Jun 2026).